All posts by Kartik Srinivas
The application of computational mechanics analysis techniques to elastomers presents unique challenges in modeling the following characteristics:
– The load-deflection behaviour of an elastomer is markedly non-linear.
– The recoverable strains can be as high 400 % making it imperative to use the large
– The stress-strain characteristics are highly dependent on temperature and rate effects are pronounced.
– Elastomers are nearly incompressible.
– Viscoelastic effects are significant.
The ability to model the special elastomer characteristics requires the use of sophisticated material models and non-linear Finite element analysis tools that are different in scope and theory than those used for metal analysis. Elastomers also call for superior analysis methodologies as elastomers are generally located in a system comprising of metal-elastomer parts giving rise to contact-impact and complex boundary conditions. The presence of these conditions require a judicious use of the available element technology and solution techniques.
FEA Support Testing
Most commercial FEA software packages use a curve-fitting procedure to generate the material constants for the selected material model. The input to the curve-fitting procedure is the stress-strain or stress-stretch data from the following physical tests:
1 Uniaxial tension test
2 Uniaxial compression test OR Equibiaxial tension test
3 Planar shear test
4 Volumetric compression test
A minimum of one test data is necessary, however greater the amount of test data, better the quality of the material constants and the resulting simulation. Testing should be carried out for the deformation modes the elastomer part may experience during its service life.
The stress-strain data from the FEA support tests is used in generating the material constants using a curve-fitting procedure. The constants are obtained by comparing the stress-strain results obtained from the material model to the stress-strain data from experimental tests. Iterative procedure using least-squares fit method is used to obtain the constants, which reduces the relative error between the predicted and experimental values. The linear least squares fit method is used for material models that are linear in their coefficients e.g Neo-Hookean, Mooney-Rivlin, Yeoh etc. For material models that are nonlinear in the coefficient relations e.g. Ogden etc, a nonlinear least squares method is used.
Verification and Validation
In the FEA of elastomeric components it is necessary to carry out checks and verification steps through out the analysis. The verification of the material model and geometry can be carried out in three steps,
_ Initially a single element test can be carried out to study the suitability of the chosen material model.
_ FE analysis of a tension or compression support test can be carried out to study the material characteristics.
_ Based upon the feedback from the first two steps, a verification of the FEA model
can be carried out by applying the main deformation mode on the actual component
on any suitable testing machine and verifying the results computationally.
Figure(1) shows the single element test for an elastomeric element, a displacement
boundary condition is applied on a face, while constraining the movement of the opposite face. Plots A and B show the deformed and undeformed plots for the single element. The load vs. displacement values are then compared to the data obtained from the experimental tests to judge the accuracy of the hyperelastic material model used.
Figure (2) shows the verification procedure carrying out using an FEA support test.
Figure shows an axisymmetric model of the compression button. Similar to the single
element test, the load-displacement values from the Finite element analysis are compared to the experimental results to check for validity and accuracy. It is possible that the results may match up very well for the single element test but may be off for the FEA support test verification by a margin. Plot C shows the specimen in a testing jig. Plot D and E show the undeformed and deformed shape of the specimen.
Figure(3) shows the verification procedure that can be carried out to verify the FEA
Model as well as the used material model. The procedure also validates the boundary conditions if the main deformation mode is simulated on an testing machine and results verified computationally. Plot F shows a bushing on a testing jig, plots G and H show the FEA model and load vs. displacement results compared to the experimental results. It is generally observed that verification procedures work very well for plane strain and axisymmetric cases and the use of 3-D modeling in the present procedure provides a more rigorous verification methodology.
AdvanSES provides Hyperelastic, Viscoelastic Material Characterization Testing for CAE & FEA softwares.
Unaged and Aged Properties and FEA Material Constants for all types of Polymers and Composites. Mooney-Rivlin, Ogden, Arruda-Boyce, Blatz-ko, Yeoh, Polynomials etc.
OEMs and Tier 1 and tier 2 suppliers are under increased pressure to meet with 5 years and 100 thousand miles warranty on each and every rubber and plastic component in a passenger vehicle. With customer expectations for 100% reliability there is a need to fully test all the design, prototyping stage . Rubber mounts and vibration isolators are used to connect a vehicle’s engine and transmission to its chassis.
They are required to reduce the transmission of vibrations, providing the desired ride characteristics and protecting the rigid frame. The elastomeric material used in these components tends to naturally degrade over a period of time due to harsh conditions in addition to material degradation. Testing the properties of these mounts under such conditions is critical to make sure that the components meet the expectations. Choosing the correct test parameters required to perform this type of automotive component tests reliably can be challenging. We have more than a decade of experience in the design, analysis and testing of these rubber elastomer mounts.
The basic material properties that need to be known for accurate Finite Element Analysis are:
1) Stress-Strain relationship, Young’s modulus of elasticity
2) Yield strength.
3) Bulk and shear modulus.
4) Poisson’s ratio.
The relationship between applied mechanical force/stress and resulting deformation/strain on a given material is called its stress-strain curve or a constituent relationship equation. Isotropic materials like plastics, metals and elastomers exhibit the same behavior in all their orientations and layouts, while orthotropic and anisotropic materials, such as composites, glass-filled plastics, etc. show mechanical stress-strain properties that vary depending on the directions. Stress-strain equations and behaviors vary greatly by material type, as metals, plastics, elastomers, glass, composites, etc.
AdvanSES is a premier material testing laboratory for mechanical properties testing and component testing. We can test aged and unaged samples. Materials can be tested under static, dynamic and under elevated temperature conditions. In addition to testing and FEA, we have in-house machining facilities to develop test fixtures required to evaluate products and components for stiffness, stress, strain etc.
Non-linear Viscoelastic Dynamic Properties of Polymer, Rubber and Elastomer Materials
Static testing of materials as per ASTM D412, ASTM D638, ASTM D624 etc can be cate- gorized as slow speed tests or static tests. The difference between a static test and dynamic test is not only simply based on the speed of the test but also on other test variables em- ployed like forcing functions, displacement amplitudes, and strain cycles. The difference is also in the nature of the information we back out from the tests. When related to poly- mers and elastomers, the information from a conventional test is usually related to quality control aspect of the material or the product, while from dynamic tests we back out data regarding the functional performance of the material and the product.
Tires are subjected to high cyclical deformations when vehicles are running on the road. When exposed to harsh road conditions, the service lifetime of the tires is jeopardized by many factors, such as the wear of the tread, the heat generated by friction, rubber aging, and others. As a result, tires usually have composite layer structures made of carbon-filled rubber, nylon cords, and steel wires, etc. In particular, the composition of rubber at different layers of the tire architecture is optimized to provide different functional properties. The desired functionality of the different tire layers is achieved by the strategical design of specific viscoelastic properties in the different layers. Zones of high loss modulus material will absorb energy differently than zones of low loss modulus. The development of tires utilizing dynamic characterization allows one to develop tires for smoother and safer rides in different weather conditions.
Figure Locations of Different Materials in a Tire Design
The dynamic properties are also related to tire performance like rolling resistance, wet traction, dry traction, winter performance and wear. Evaluation of viscoelastic properties of different layers of the tire by DMA tests is necessary and essential to predict the dynamic performance. The complex modulus and mechanical behavior of the tire are mapped across the cross section of the tire comprising of the different materials. A DMA frequency sweep
test is performed on the tire sample to investigate the effect of the cyclic stress/strain fre- quency on the complex modulus and dynamic modulus of the tire, which represents the viscoelastic properties of the tire rotating at different speeds. Significant work on effects of dynamic properties on tire performance has been carried out by Ed Terrill et al. at Akron Rubber Development Laboratory, Inc.
Non-linear Viscoelastic Tire Simulation Using FEA
Non-linear Viscoelastic tire simulation is carried out using Abaqus to predict the hysteresis losses, temperature distribution and rolling resistance of a tire. The simulation includes several steps like (a) FE tire model generation, (b) Material parameter identification, (c) Material modeling and (d) Tire Rolling Simulation. The energy dissipation and rolling re- sistance are evaluated by using dynamic mechanical properties like storage and loss modu- lus, tan delta etc. The heat dissipation energy is calculated by taking the product of elastic strain energy and the loss tangent of materials. Computation of tire rolling is further carried out. The total energy loss per one tire revolution is calculated by;
Ψdiss = ∑ i2πΨiTanδi, (.27)
where Ψ is the elastic strain energy,
Ψdiss is the dissipated energy in one full rotation of the tire, and
Tanδi, is the damping coefficient.
The temperature prediction in a rolling tire shown in Fig (2) is calculated from the loss modulus and the strain in the element at that location. With the change in the deformation pattern, the strains are also modified in the algorithm to predict change in the temperature distribution in the different tire regions.
Rubber components under multiaxial cyclic loading conditions are often considered to have failed or degraded when there is a change is the stiffness of the component and it is no longer able to provide the performance it is designed for. Elastomeric polymer components are widely used in many industries like automotive, aerospace and biomedical applications due to their good vibration isolation and energy absorption characteristics. The type of loading normally encountered by these components in service is mutiaxial in nature. Fatigue failure is thus a major consideration in their design and availability of testing techniques to predict fatigue life under these complex conditions is a necessity.
In real world applications all materials and products are subjected to a wide variety of vibrating or oscillating forces. Fatigue testing consists of applying a cyclic load to a test specimen or the component to determine in-service performance during situations similar to real world working conditions.
Advanced Scientific and Engineering Services (AdvanSES) specializes in Fatigue Testing of Automotive Components including hoses, engine mounts, vibration isolators, silent bushes etc. Fatigue testing can be carried out in stress & force control, strain control or displacement control. The deformation modes under which fatigue tests are generally carried out are tension – tension, compression – compression, tension – compression and compression – tension.
ASTM D 5992 Test Standard applies to Dynamic Properties of Rubber Vibration Products such as springs, dampers, and flexible load-carrying devices, flexible power transmission couplings, vibration isolation components and mechanical rubber goods. The standard applies to to the measurement of stiffness, damping, and measurement of dynamic modulus.
Dynamic testing is performed on a variety of rubber parts and components like engine mounts, hoses, conveyor belts, vibration isolators, laminated and non-laminated bearing pads, silent bushes etc. to determine their response to dynamic loads and cyclic loading.
Personalized consultation from AdvanSES engineers can streamline testing and provide the necessary tools and techniques to accurately evaluate material performance under field service conditions.
The quantities of interest for measurements are tan delta, loss modulus, storage modulus, phase etc. All of these properties are viscoleastic properties and require instruments, techniques and measurement practices of the highest quality.
ASTM D5992 and ISO 4664-1
ASTM D5992 covers the methods and process available for determining the dynamic prop- erties of vulcanized natural rubber and synthetic rubber compounds and components. The standard covers the sample shape and size requirements, the test methods, and the pro- cedures to generate the test results data and carry out further subsequent analysis. The methods described are primarily useful over the range of temperatures from cryogenic to 200◦C and for frequencies from 0.01 to 100 Hz, as not all instruments and methods will accommodate the entire ranges possible for material behavior.
Figures(.43and.44) show the results from a frequency sweep test on five (5) different elastomer compounds. Results of Storage modulus and Tan delta are plotted.
Figure .43: Plot of Storage Modulus Vs Frequency from a Frequency Sweep Test
The frequency sweep tests have been carried out by applying a pre-compression of 10 % and subsequently a displacement amplitude of 1 % has been applied in the positive and negative directions. Apart from tests on cylindrical and square block samples ASTM D5992 recommends the dual lap shear test specimen in rectangular, square and cylindri- cal shape specimens. Figure (.45) shows the double lap shear shapes recommended in the standard.
Figure .44: Plot of Tan delta Vs Frequency from a Frequency Sweep Test
Figure .45: Double Lap Shear Shapes
My short book on Dynamic Properties of Polymer Materials and Their Measurements provides an introduction to solid polymer materials and their time, frequency and temperature dependent properties. It provides a review of the basic theory, how the materials are experimentally characterized, and how to analyze and predict their behavior under different service conditions.
Dynamic Properties of Polymer Materials and their Measurements
Polymer materials in their basic form exhibit a range of characteristics and behavior from elastic solid to a viscous liquid. These behavior and properties depend on the temperature, frequency and time scale at which the material or the engineering component is analyzed.
The viscous liquid polymer is defined as by having no definite shape and flow deformation under the effect of applied load is irreversible. Elastic materials such as steels and aluminium deform instantaneously under the application of load and return to the original
state upon the removal of load, provided the applied load is within the yield or plastic limits of the material. An elastic solid polymer is characterized by having a definite shape that deforms under external forces, storing this deformation energy and giving it back upon
the removal of applied load. Material behavior which combines both viscous liquid and solid like features is termed as Viscoelasticity. These viscoelastic materials exhibit a time dependent behavior where the applied load does not cause an instantaneous deformation,
but there is a time lag between the application of load and the resulting deformation. We also observe that in polymeric materials the resultant deformation also depends upon the speed of the applied load.
Characterization of dynamic properties play an important part in comparing mechanical properties of different polymers for quality, failure analysis and new material qualification. Figures 1.4 and 1.5 show the responses of purely elastic, purely viscous and of a viscoelastic material. In the case of purely elastic, the stress and the strain (force and resultant deformation) are in perfect sync with each other, resulting in a phase angle of 0. For a purely viscous response the input and resultant deformation are out of phase by 90o. For a
viscoleastic material the phase angle lies between 0 and 90 degree. Generally the measurements of viscoelastic materials are represented as a complex modulus E* to capture both viscous and elastic behavior of the material. The stress is the sum of an in-phase response and out-of-phase responses.
The so x Cosdelta term is in phase with the strain, while the term so x Sindelta is out of phase with the applied strain. The modulus E’ is in phase with strain while, E” is out of phase with the strain. The E’ is termed as storage modulus, and E” is termed as the loss modulus.
E’ = s0 x cosdelta
E” = s0 x sindelta
Tan delta = Loss factor = E”/E’